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Proof of E = mc^2?

Date: 05/04/2001 at 09:38:26
From: John
Subject: Proof of E = mc^2


I'm looking for a mathematical proof of the equation E = mc^2. I 
noticed in your archives that you talk about some of the implications 
of this equation, but don't show the line-by-line proof of it. I think 
I've almost got it, but maybe you can help me finish it off. See 

So far I have: 

An object that emits a ray of light will recoil at a velocity v, given 

     v = EM   ....................[1]

where E is the energy of the emission and M is the mass of the object 
emitting the light. 

Also, the recoil distance x, of the object, will be given by:

     x = vt   ....................[2]

where v is the velocity of the object and t is time. 

Assuming that the beam travels a distance L, and because the light 
beam moves at the speed of light, the time t, taken for the beam to 
travel across L, is given by:

     t = L/c   ...................[3]

where c is the speed of light.

Substituting [1] and [3] into [2] gives:

     x = (E/M).(L/c)   ...........[4]

Now because the center of mass doesn't move, we can say:

     Mx = mL   ...................[5]

(but I'm not sure why this is, although I know that m represents the 
apparent mass of the light).

Now by combining [4] and [5] you should be able to get E = Mc^2, but I 
just can't seem to get it!

Please help.

Date: 05/06/2001 at 16:46:49
From: Doctor Mitteldorf
Subject: Re: Proof of E = mc^2

Dear John,

I'm afraid that there is no proof. E = mc^2 is a fact about nature, 
not about mathematics. It was a leap of insight on Einstein's part, 
and physicists believe it to be true not because Einstein proved it 
but because, with E = mc^2 as a foundation, he was able to construct a 
theory that predicts so many things correctly.

There are some equations from physics that can be derived from more 
fundamental equations, and in that sense can be "proved," but this is 
not one of them. It's a fundamental postulate.

The meaning of the equation is that what up until Einstein's time had 
been called "energy" and what had been called "mass" are really the 
same thing, and the equation gives a conversion for going from the 
old, conventional measures of mass to the corresponding amount of 

- Doctor Mitteldorf, The Math Forum   
Associated Topics:
High School Physics/Chemistry

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